Which relation correctly connects apparent power S with real power P and reactive power Q in a sinusoidal system?

• A. S^2 = P^2 + Q^2
• B. S = P + Q
• C. P = S cos φ
• D. Q = P sin φ

Answer: (A)

In sinusoidal steady-state, apparent power S is the magnitude of the complex power, which can be written as P + jQ. The real power P lies in phase with the voltage, while the reactive power Q is 90 degrees out of phase. These three quantities form a power triangle, where S is the vector hypotenuse. The relationship S^2 = P^2 + Q^2 comes from taking the magnitude: S = sqrt(P^2 + Q^2). This also ties to the angle φ between voltage and current, with P = S cos φ and Q = S sin φ, so dividing gives tan φ = Q/P. Therefore, the direct connection among all three is S^2 = P^2 + Q^2 (and equivalently S = sqrt(P^2 + Q^2)). S = P + Q would ignore the vector nature of the powers; Q = P sin φ is not correct (the correct forms are Q = S sin φ or Q = P tan φ).